A one-parametric class of merit functions for the symmetric cone complementarity problem

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

A one-parametric class of merit functions for the symmetric cone complementarity problem

Shaohua Pan

^a,1

, Jein-Shan Chen

^b,∗,2

aSchool of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China bDepartment of Mathematics, National Taiwan Normal University, Taipei, Taiwan 11677

a r t i c l e i n f o a b s t r a c t

Article history:

Received 23 June 2008 Available online 5 February 2009 Submitted by H. Frankowska

Keywords:

Symmetric cone complementarity problem Merit function

Jordan algebra Smoothness Lipschitz continuity Cartesian P -properties

In this paper, we extend the one-parametric class of merit functions proposed by Kan- zow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton- type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227–251] for the nonnegative orthant complementarity problem to the general symmet- ric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer–Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Ya- mashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl.

97 (1997) 115–135] to the SCCP. By exploiting the Cartesian P -properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasi- ble point or the SCCP with the joint Cartesian R02-property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimiza- tion reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293–327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradi- ent of the squared norm of the matrix-valued Fischer–Burmeister function, Math. Program.

107 (2006) 547–553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159–185] under a unified framework.

©2009 Elsevier Inc. All rights reserved.

1. Introduction

Given a Euclidean Jordan algebra

A = (V, ◦, ·,·)

^where

V

is a finite-dimensional vector space over the real field

R

endowed with the inner product

·,·

^{and “}

◦

” denotes the Jordan product. Let

K

be a symmetric cone in

V

^{and G}

,

F

: V → V

be nonlinear transformations assumed to be continuously differentiable throughout this paper. Consider the symmetric cone complementarity problem (SCCP) of finding

ζ ∈ V

^{such that}

G

(ζ ) ∈ K,

F

(ζ ) ∈ K, 

G

(ζ ),

F

(ζ ) 

=

⁰

.

(1)

*

Corresponding author.

E-mail addresses:shhpan@scut.edu.cn(S. Pan),jschen@math.ntnu.edu.tw(J.-S. Chen).

1 Partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.

2 Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. Partially supported by National Science Council of Taiwan.

0022-247X/$ – see front matter ©2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmaa.2009.01.064

The model provides a simple, natural and unified framework for various existing complementarity problems such as the nonnegative orthant nonlinear complementarity problem (NCP), the second-order cone complementarity problem (SOCCP), and the semidefinite complementarity problem (SDCP). In addition, the model itself is closely related to the KKT optimality conditions for the convex symmetric cone program (CSCP):

minimize g

(

x

),

subject to



^ai

,

x

 =

^bi

,

i

=

¹

,

2

, . . . ,

m

,

x

∈ K,

⁽²⁾

where a_i

∈ V

^{, b}i

∈ R

^{for i}

=

¹

,

2

, . . . ,

m, and g

: V → R

is a convex twice continuously differentiable function. Therefore, the SCCP has wide applications in engineering, economics, management science and other fields; see [1,11,20,29] and references therein.

During the past several years, interior-point methods have been well used for solving the symmetric cone linear pro- gramming problem (SCLP), i.e., the CSCP with g being a linear function (see [7,8,23,24]). However, in view of the wide applications of the SCCP, it is worthwhile to explore other solution methods for the more general CSCP and SCCP. Recently, motivated by the successful applications of the merit function approach in the solution of NCPs, SOCCPs and SDCPs (see, e.g., [4,10,22,28]), some researchers started with the investigation of merit functions or complementarity functions associ- ated with symmetric cones. For example, Liu, Zhang and Wang [21] extended a class of merit functions proposed in [18] to the following special SCCP:

ζ ∈ K,

^F

(ζ ) ∈ K,  ζ,

F

(ζ ) 

=

⁰

;

⁽³⁾

Kong, Tuncel and Xiu [17] studied the extension of the implicit Lagrangian function proposed by Mangasarian and Solodov [22] to symmetric cones; and Kong, Sun and Xiu [16] proposed a regularized smoothing method for the SCCP (3) based on the natural residual complementarity function associated with symmetric cones. Following this line, in this paper we will consider the extension of the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [14]

and a class of regularized functions based on it.

We define the one-parametric class of vector-valued functions

φ

_τ

: V × V → V

^by

φ

τ

(

x

,

y

) := 

x²

+

^y2

+ ( τ −

²

)(

x

◦

^y

) 

1/2

− (

^x

+

^y

),

(4)

where

τ _{∈ (}

0

,

4

)

is an arbitrary but fixed parameter, x²

=

^x

◦

^{x, x1/2} is a vector such that

(

x^1/2

)

²

=

^{x, and x}

+

y means the usual componentwise addition of vectors. When

τ ₌

2,

φ

τ reduces to the vector-valued Fischer–Burmeister function given by

φ

FB

(

x

,

y

) := 

x²

+

^y2



1/2

− (

^x

+

^y

) ;

⁽⁵⁾

whereas as

τ →

0 it will become a multiple of the vector-valued residual function

ψ

NR

(

x

,

y

) :=

^x

− (

^x

−

^y

)

₊

,

where

( ·)

+ denotes the metric projection on

K

. In this sense, the one-parametric class of vector-valued functions covers the two popular complementarity functions associated with the symmetric cone

K

. In fact, from Lemma 3.1 later, it follows that the function

φ

τ with any

τ ∈ (

⁰

,

4

)

is a complementarity function associated with

K

^{, that is,}

φ

_τ

(

x

,

y

) =

⁰

⇐⇒

^x

∈ K,

^y

∈ K, 

^x

,

y

 =

⁰

.

Consequently, its squared norm yields a merit function associated with

K ψ

_τ

(

x

,

y

) :=

¹

2

φ

_τ

(

x

,

y

)

²

,

(6)

where is the norm induced by

·,·

, and the SCCP can be reformulated as min

ζ∈V fτ

(ζ ) := ψ

τ



G

(ζ ),

F

(ζ ) 

.

(7)

To apply the effective unconstrained optimization methods, such as the quasi-Newton method, the trust-region method and the conjugate gradient method, for solving the unconstrained minimization reformulation (7) of the SCCP, the smooth- ness of the merit function

ψ

τ and the Lipschitz continuity of its gradient will play an important role. In Sections 3 and 4, we show that the function

ψ

τ defined by (6) is continuously differentiable everywhere and has a globally Lipschitz continuous gradient with the Lipschitz constant being a positive multiple of 1

+ τ

⁻¹. These results generalize some recent important works in [4,25,28] under a unified framework, as well as improve the work [21] greatly in which only the differentiability of the merit function

ψ

FB was given.

In addition, we also consider a class of regularized functions for fτ defined as

ˆ

_fτ

(ζ ) := ψ

0



G

(ζ ) ◦

^F

(ζ )  + ψ

τ



G

(ζ ),

F

(ζ ) 

,

(8)

where

ψ

0

: V → R

+ is continuously differentiable and satisfies

ψ

0

(

u

) =

⁰

∀

^u

∈ −K

^and

ψ

0

(

u

)  β  (

u

)

₊

 ∀

^u

∈ V

⁽⁹⁾

for some constant

β >

0. Using the properties of

ψ

0 in (9), it is not hard to verify that

ˆ

fτ is a merit function for the SCCP.

The class of functions will reduce to the one studied in [21] if

τ =

2 and G degenerates into an identity transformation. In Section 5, we show that the class of merit functions can provide a global error bound for the solution of the SCCP under the condition that G and F have the joint uniform Cartesian P -property. In Section 6, we establish the boundedness of the level sets of

ˆ

fτ under a weaker condition than the one used by [21], which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with G and F having the joint Cartesian R02-property.

Throughout this paper,

I

denotes an identity operator, represents the norm induced by the inner product

·,·

^{, and} int

( K)

denotes the interior of the symmetric cone

K

. All vectors are column ones and write the column vector

(

x₁^T

, . . . ,

x_m^T

)

^T as

(

x1

, . . . ,

xm

)

, where xi is a column vector from the subspace

V

i. For any x

∈ V

, we denote

(

x

)

₊ and

(

x

)

₋ by the metric projection of x onto

K

^and

−K

, respectively, i.e.,

(

x

)

₊

:=

^{arg min}y∈K

{

^x

−

^y

}

. For any symmetric matrix A, the notation A



O means that A is positive semidefinite. For a differentiable mapping F

: V → V

, the notation

∇

^F

(

x

)

denotes the transposed Jacobian operator of F at a point x. We write x

=

^o

( α )

(respectively, x

=

^O

( α )

) if

^x

/| α _{| →}

0 (respectively, uniformly bounded) as

α _→

0.

2. Preliminaries

In this section, we recall some concepts and materials of Euclidean Jordan algebras that will be used in the subsequent sections. More detailed expositions of Euclidean Jordan algebras can be found in the monograph by Faraut and Korányi [9].

Besides, one can find excellent summaries in the articles [2,12,24,26].

A Euclidean Jordan algebra is a triple

(V, ◦, ·,·

V

)

, where

V

is a finite-dimensional inner product space over the real field

R

^and

(

x

,

y

) →

^x

◦

^y

: V × V → V

is a bilinear mapping satisfying the following conditions:

(i) x

◦

^y

=

^y

◦

x for all x

,

y

∈ V

^,

(ii) x

◦ (

^x2

◦

^y

) =

^x2

◦ (

^x

◦

^y

)

for all x

,

y

∈ V

^{, where x2}

:=

^x

◦

^{x, and} (iii)



^x

◦

^y

,

z



V

= 

^x

,

y

◦

^z



V for all x

,

y

,

z

∈ V

^.

We call x

◦

y the Jordan product of x and y. We also assume that there is an element e

∈ V

, called the unit element, such that x

◦

^e

=

x for all x

∈ V

^{. For x}

∈ V

^{, let}

ζ (

x

)

be the degree of the minimal polynomial of x, which can be equivalently defined as

ζ (

x

) :=

^min



k:



e

,

x

,

x²

, . . . ,

x^k



are linearly dependent

 .

Since

ζ (

x

) 

^dim

( V)

^{where dim}

( V)

denotes the dimension of

V

, the rank of

V

is well defined by r

:=

^max

{ζ(

^x

)

: x

∈ V}

^{. In a} Euclidean Jordan algebra

A = (V, ◦, ·,·

V

)

, we define the set of squares as

K := {

x²: x

∈ V}

. Then, by Theorem III.2.1 of [9],

K

is a symmetric cone. This means that

K

is a self-dual closed convex cone with nonempty interior int

(K)

and for any two elements x

,

y

∈

^int

(K)

, there exists an invertible linear transformation

T : V → V

^{such that}

T (K) = K

^and

T (

^x

) =

^y.

A Euclidean Jordan algebra is said to be simple if it is not the direct sum of two Euclidean Jordan algebras. By Propo- sition III.4.4 of [9], each Euclidean Jordan algebra is, in a unique way, a direct sum of simple Euclidean Jordan algebras.

A common simple Euclidean Jordan algebra is

( S

ⁿ

, ◦, ·,·

Sⁿ

)

, where

S

ⁿ is the space of n

×

n real symmetric matrices with the inner product



^X

,

Y



Sⁿ

:=

^Tr

(

X Y

)

, and the Jordan product is defined by X

◦

^Y

:= (

^{X Y}

+

^{Y X}

)/

2. Here, X Y is the usual matrix multiplication of X and Y and Tr

(

X

)

is the trace of X . The associate cone

K

is the set of all positive semidef- inite matrices. Another one is the Lorentz algebra

( R

ⁿ

, ◦, ·,·

Rⁿ

)

, where

R

ⁿ is the Euclidean space of dimension n with the standard inner product



^x

,

y



Rⁿ

=

^xTy, and the Jordan product is defined by x

◦

^y

:= (

^x

,

y



Rⁿ

,

x1y2

+

^y1x2

)

for any x

= (

^x1

,

x2

),

y

= (

^y1

,

y2

) ∈ R × R

ⁿ⁻¹. The associate cone, called the Lorentz cone or the second-order cone, is

K := 

x

= (

^x1

,

x2

) ∈ R × R

^n−1:

^x2



^x1

 .

Recall that an element c

∈ V

is said to be idempotent if c²

=

c. Two idempotents c and d are said to be orthogonal if c

◦

^d

=

0. One says that

{

^c1

,

c2

, . . . ,

ck

}

is a complete system of orthogonal idempotents if

c²_j

=

^cj

,

c_j

◦

^ci

=

^{0 if j}

=

ⁱ

,

j

,

i

=

¹

,

2

, . . . ,

k

,

and

k

j=¹ c_j

=

^e

.

A nonzero idempotent is said to be primitive if it cannot be written as the sum of two other nonzero idempotents. We call a complete system of orthogonal primitive idempotents a Jordan frame. Then, we have the following spectral decomposition theorem.

Theorem 2.1. (See [9, Theorem III.1.2].) Suppose that

A = (V, ◦, ·,·

V

)

is a Euclidean Jordan algebra and the rank of

A

^{is r. Then} for any x

∈ V

, there exist a Jordan frame

{

^c1

,

c₂

, . . . ,

c_r

}

and real numbers

λ

1

(

x

), λ

2

(

x

), . . . , λ

r

(

x

)

, arranged in the decreasing order

λ

1

(

x

)  λ

2

(

x

)  · · ·  λ

r

(

x

)

, such that x

=

r

j=¹

λ

j

(

x

)

cj.

The numbers

λ

j

(

x

)

(counting multiplicities), which are uniquely determined by x, are called the eigenvalue, and we write the maximum eigenvalue and the minimum eigenvalue of x as

λ

max

(

x

)

and

λ

min

(

x

)

, respectively. The trace of x, denoted as tr

(

x

)

, is defined by tr

(

x

) :=

_r

j=¹

λ

j

(

x

)

; whereas the determinant of x is defined by det

(

x

) := 

_r

j=¹

λ

j

(

x

)

.

By Proposition III.1.5 of [9], a Jordan algebra over

R

with a unit element e

∈ V

is Euclidean if and only if the symmetric bilinear form tr

(

x

◦

^y

)

is positive definite. Hence, we may define an inner product

·,·

^on

V

^by



^x

,

y

 :=

^tr

(

x

◦

^y

), ∀

^x

,

y

∈ V.

⁽¹⁰⁾

Unless otherwise states, the inner product

·,·

appearing in this paper always means the one defined by (10). By the asso- ciativity of tr

( ·)

(see [9, Proposition II.4.3]), the inner product

·,·

is associative, i.e.,



^x

,

y

◦

^z

 = 

^y

,

x

◦

^z



^{for all x}

,

y

,

z

∈ V

^. Let

L(

x

)

y

:=

^x

◦

^y for every y

∈ V.

Then, the linear operator

L(

^x

)

for each x

∈ V

is symmetric with respect to the inner product

·,·

in the sense that

L(

^x

)

y

,

z

 = 

^y

, L(

^x

)

z



^{for any y}

,

z

∈ V

^{. Let} be the norm on

V

induced by the inner product

·,·

^{, namely,}

^x

:= 



^x

,

x

 =

_r

j=1

λ

²_j

(

x

)



1/2

, ∀

^x

∈ V.

It is not difficult to verify that for any x

,

y

∈ V

, there always holds that



^x

,

y

 

¹ 2



^x

²

+

^y

²



and

^x

+

^y

²



²



^x

²

+

^y

²



.

(11)

Let

ϕ _{: R → R}

be a scalar valued function. Then, it is natural to define a vector-valued function associated with the Euclidean Jordan algebra

A = (V, ◦, ·,·)

^by

ϕ

_V

(

x

) := ϕ ^ λ

1

(

x

) 

c1

+ ϕ ^ λ

2

(

x

) 

c2

+ · · · + ϕ ^ λ

r

(

x

) 

cr

,

(12)

where x

∈ V

has the spectral decomposition x

=

r

j=¹

λ

j

(

x

)

cj. The function

ϕ

_V is also called the Löwner operator [26].

When

ϕ (

t

)

is chosen as max

{

⁰

,

t

}

^{and min}

{

⁰

,

t

}

^{for t}

∈ R

, respectively,

ϕ

_V becomes the metric projection operator onto

K

and

−K

^:

(

x

)

₊

:=

r j=1

max



0

, λ

j

(

x

) 

cj and

(

x

)

₋

:=

r j=1

min



0

, λ

j

(

x

) 

cj

.

(13)

Lemma 2.1. (See [26, Theorem 13].) For any x

=

r

j=¹

λ

j

(

x

)

cj, let

ϕ

_Vbe given as in (12). Then

ϕ

_Vis (continuously) differentiable at x if and only if

ϕ

is (continuously) differentiable at each

λ

j

(

x

)

, j

=

¹

,

2

, . . . ,

r. The derivative of

ϕ

_Vat x, for any h

∈ V

^{, is}

ϕ

^_V

(

x

)

h

=

r

j=¹

 ϕ

^[1]

^ λ(

x

) 

j j



^cj

,

h



^cj

+

1^j<l^r 4



ϕ

^[1]

^ λ(

x

) 

jlcj

◦ (

^cl

◦

^h

),

where

 ϕ

^[1]

^ λ(

x

) 

i j

:=



_{ϕ(λi(x))−ϕ(λj(x))}

λi(x)−λj(x) if

λ

i

(

x

) = λ

j

(

x

), ϕ

^

(λ

i

(

x

))

if

λ

i

(

x

) = λ

j

(

x

),

i

,

j

=

¹

,

2

, . . . ,

r

.

In fact, the Jacobian

ϕ

_V^

( ·)

is a linear and symmetric operator, which can be written as

ϕ

^_V

(

x

) =

r

j=¹

ϕ

^

^ λ

j

(

x

) 

Q(

^cj

) +

²

r i,j=¹,i=^j

 ϕ

^[1]

^ λ(

x

) 

i j

L(

^cj

)L(

^ci

),

(14)

where

Q(

^x

) :=

²

L

²

(

x

) − L(

^x2

)

for any x

∈ V

is called the quadratic representation of

V

^.

In the sequel, unless otherwise stated, we assume that

A = (V, ◦, ·,·)

is a simple Euclidean Jordan algebra of rank r and dim

( V) =

^n.

An important part in the theory of Euclidean Jordan algebras is the Peirce decomposition. Let c be a nonzero idempotent in

A

. Then, by [9, Proposition III.1.3], c satisfies 2

L

³

(

c

) −

³

L

²

(

c

) + L(

^c

) =

0 and the distinct eigenvalues of the symmetric operator

L(

^c

)

are 0, ¹₂ and 1. Let

V(

^c

,

1

), V(

^c

,

¹₂

)

and

V(

^c

,

0

)

be the three corresponding eigenspaces, i.e.,

V(

^c

, α ) := 

x

∈ V

^:

L(

^c

)

x

= α

x



, α =

¹

,

¹ 2

,

0

.

Then

V

is the orthogonal direct sum of

V(

^c

,

1

)

,

V(

^c

,

¹₂

)

and

V(

^c

,

0

)

. The decomposition

V = V(

^c

,

1

) ⊕ V



c

,

¹

2



⊕ V(

^c

,

0

)

is called the Peirce decomposition of

V

with respect to the nonzero idempotent c.

Let

{

^c1

,

c2

, . . . ,

cr

}

be a Jordan frame of

A

^{. For i}

,

j

∈ {

¹

, . . . ,

r

}

, define the eigenspaces

V

ii

:= V(

^ci

,

1

) = R

^ci

,

V

i j

:= V



c_i

,

¹

2



∩ V



c_j

,

¹

2

 ,

i

=

^j

.

Then, from [9, Theorem IV.2.1], it follows that the following conclusion holds.

Theorem 2.2. The space

V

is the orthogonal direct sum of subspaces

V

i j

(

1



ⁱ



^j



^r

)

, i.e.,

V = 

i^jVi j. Furthermore,

V

i j

◦ V

i j

⊂ V

ii

+ V

j j

, V

i j

◦ V

jk

⊂ V

ik

,

if i

=

^k

,

V

i j

◦ V

kl

= {

⁰

},

^if

{

ⁱ

,

j

} ∩ {

^k

,

l

} = ∅.

Let x

∈ V

have the spectral decomposition x

=

r

j=1

λ

j

(

x

)

cj. For i

,

j

∈ {

¹

,

2

, . . . ,

r

}

^{, let}

C

i j

(

x

)

be the orthogonal projection operator onto

V

i j. Then,

C

i j

(

x

) = C

i j^∗

(

x

), C

²i j

(

x

) = C

i j

(

x

), C

i j

(

x

)C

kl

(

x

) =

^{0 if}

{

ⁱ

,

j

} = {

^k

,

l

},

ⁱ

,

j

,

k

,

l

=

¹

, . . . ,

r

,

(15) and

1ijr

C

i j

(

x

) = I,

⁽¹⁶⁾

where

C

_{i j}^∗ is the adjoint (operator) of

C

i j. In addition, by [9, Theorem IV.2.1],

C

j j

(

x

) = Q(

^cj

)

and

C

i j

(

x

) =

⁴

L(

^ci

) L(

^cj

) =

⁴

L(

^cj

) L(

^ci

) = C

ji

(

x

),

i

,

j

=

¹

,

2

, . . . ,

r

.

Note that the original notation in [9] for orthogonal projection operator is Pi j. However, to avoid confusion with another orthogonal projector Pi

(

cj

)

onto

V(

^c

, α )

and orthogonal matrix P which will be used later (Sections 3 and 4), we adopt

C

i j instead.

With the orthogonal projection operators

{C

i j

(

x

)

: i

,

j

=

¹

,

2

, . . . ,

r

}

, we have the following spectral decomposition theo- rem for

L(

^x

)

and

L(

^x2

)

; see [15, Chapters VI–V].

Lemma 2.2. Let x

∈ V

have the spectral decomposition x

=

_r

j=1

λ

j

(

x

)

c_j. Then the symmetric operator

L(

^x

)

has the spectral decom- position

L(

^x

) =

r

j=1

λ

j

(

x

) C

j j

(

x

) +

1^j<l^r 1 2

 λ

j

(

x

) + λ

l

(

x

)  C

jl

(

x

)

with the spectrum

σ ( L(

^x

))

consisting of all distinct numbers in

{

¹₂

(λ

j

(

x

) + λ

l

(

x

))

: j

,

l

=

¹

,

2

, . . . ,

r

}

^{, and}

L(

^x2

)

has the spectral decomposition

L 

x²



=

r

j=¹

λ

²_j

(

x

)C

j j

(

x

) +

1^j<l^r 1 2

 λ

²_j

(

x

) + λ

²_l

(

x

) 

C

jl

(

x

)

(17)

with the spectrum

σ (L(

^x2

))

consisting of all distinct numbers in

{

¹₂

(λ

²_j

(

x

) + λ

²_l

(

x

))

: j

,

l

=

¹

,

2

, . . . ,

r

}

^. Proposition 2.1. For any x

∈ V

, the operator

L(

^x2

) − L

²

(

x

)

is positive semidefinite.

Proof. By Lemma 2.2 and (15), we can verify that

L

²

(

x

)

has the spectral decomposition

L

²

(

x

) =

r j=¹

λ

²_j

(

x

)C

j j

(

x

) +

1^j<l^r 1 4

 λ

j

(

x

) + λ

l

(

x

) 

2

C

jl

(

x

).

(18)

This means that the operator

L(

^x2

) − L

²

(

x

)

has the spectral decomposition

L 

x²



− L

²

(

x

) =

1j<lr



1 2

 λ

²_j

(

x

) + λ

_l²

(

x

) 

−

¹ 4

 λ

j

(

x

) + λ

l

(

x

) 

2

 C

jl

(

x

).

Noting that the orthogonal projection operator is positive semidefinite on

V

^and

λ

²_j

(

x

) + λ

²_l

(

x

)

2

 (λ

j

(

x

) + λ

l

(

x

))

²

4 for all j

,

l

=

¹

,

2

, . . . ,

r

,

we readily obtain the conclusion from the spectral decomposition of

L(

^x2

) − L

²

(

x

)

.

2

3. Differentiability of the functionψτ

In this section, we show that

ψ

τ is a merit function associated with

K

, and moreover, it is differentiable everywhere on

V × V

. By the definition of Jordan product,

x²

+

^y2

+ ( τ ₋

2

)(

x

◦

^y

) =



x

+ τ ₋

2 2 y



2

+ τ (

4

− τ )

4 y²

=



y

+ τ ₋

2 2 x



2

+ τ (

4

− τ )

4 x²

∈ K

⁽¹⁹⁾

for any x

,

y

∈ V

, and consequently the function

φ

τ in (4) is well defined. The following lemma states that

φ

_{τ and}

ψ

_{τ is} respectively a complementarity function and a merit function associated with

K

^.

Lemma 3.1. For any x

,

y

∈ V

^{, let}

φ

τ and

ψ

τ be given by (4) and (6), respectively. Then,

ψ

_τ

(

x

,

y

) =

⁰

⇐⇒ φ

τ

(

x

,

y

) =

⁰

⇐⇒

^x

∈ K,

^y

∈ K, 

^x

,

y

 =

⁰

.

Proof. The first equivalence is clear by the definition of

ψ

τ , and we only need to prove the second equivalence. Suppose that

φ

_τ

(

x

,

y

) =

^{0. Then,}



x²

+

^y2

+ ( τ ₋

2

)(

x

◦

^y

) 

1/2

= (

^x

+

^y

).

(20)

Squaring the two sides of (20) yields that

x²

+

^y2

+ ( τ −

²

)(

x

◦

^y

) =

^x2

+

^y2

+

²

(

x

◦

^y

),

which implies x

◦

^y

=

^{0 since}

τ ∈ (

⁰

,

4

)

. Substituting x

◦

^y

=

0 into (20), we have that x

= 

x²

+

^y2



1/2

−

^{y and y}

= 

x²

+

^y2



1/2

−

^x

.

Since x²

+

^y2

∈ K

, x²

∈ K

and y²

∈ K

, from [12, Proposition 8] or [19, Corollary 9] it follows that x

,

y

∈ K

. Consequently, the necessity holds. For the other direction, suppose x

,

y

∈ K

^{and x}

◦

^y

=

^{0. Then,}

(

x

+

^y

)

²

=

^x2

+

^y2. This, together with x

◦

^y

=

0, implies that



x²

+

^y2

+ ( τ ₋

2

)(

x

◦

^y

) 

1/2

− (

^x

+

^y

) =

⁰

.

Consequently, the sufficiency follows. The proof is thus completed.

2

In what follows, we concentrate on the differentiability of the merit function

ψ

τ . For this purpose, we need the following two crucial technical lemmas.

Lemma 3.2. For any x

,

y

∈ V

^{, let u}

(

x

,

y

) := (

^x2

+

^y2

)

^1/2. Then, the function u

(

x

,

y

)

is continuously differentiable at any point

(

x

,

y

)

such that x²

+

^y2

∈

^int

(K)

. Furthermore,

∇

xu

(

x

,

y

) = L(

^x

)L

⁻¹



u

(

x

,

y

) 

and

∇

yu

(

x

,

y

) = L(

^y

)L

⁻¹



u

(

x

,

y

) 

.

(21)

Proof. The first part is due to Lemma 2.1. It remains to derive the formulas in (21). From the definition of u

(

x

,

y

)

, it follows that

u²

(

x

,

y

) =

^x2

+

^y2

, ∀

^x

,

y

∈ V.

⁽²²⁾

By the formula (14), it is easy to verify that

∇

x

(

x²

) =

²

L(

x

)

. Differentiating on both sides of (22) with respect to x then yields that

2

∇

xu

(

x

,

y

)L 

u

(

x

,

y

) 

=

²

L(

x

).

This implies that

∇

xu

(

x

,

y

) = L(

^x

) L

⁻¹

(

u

(

x

,

y

))

since, by u

(

x

,

y

) ∈

^int

( K)

^,

L(

^u

(

x

,

y

))

is positive definite on

V

. Similarly, we have that

∇

yu

(

x

,

y

) = L(

y

)L

⁻¹

(

u

(

x

,

y

))

.

2

To present another lemma, we first introduce some related notations. For any 0

=

^z

∈ K

and z

∈ /

^int

(K)

, suppose that z has the spectral decomposition z

=

_r

j=¹

λ

j

(

z

)

c_j, where

{

^c1

,

c₂

, . . . ,

c_r

}

is a Jordan frame and

λ

1

(

z

), . . . , λ

r

(

z

)

are the eigenvalues arranged in the decreasing order

λ

1

(

z

)  λ

2

(

z

)  · · ·  λ

r

(

z

) =

0. Define the index

j^∗

:=

^min



j

 λ

j

(

z

) =

⁰

,

j

=

¹

,

2

, . . . ,

r



(23) and let

cJ

:=

j^∗−¹

l=¹

c_l

.

(24)

Clearly, j^∗ and cJ are well defined since 0

=

^z

∈ K

^{and z}

∈ /

^int

( K)

^{. Since c}J is an idempotent and cJ

=

0 (otherwise z

=

^0),

V

can be decomposed as the orthogonal direct sum of the subspaces

V(

^cJ

,

1

)

,

V(

^cJ

,

¹₂

)

and

V(

^cJ

,

0

)

. In the sequel, we write P1

(

cJ

)

, P1

2

(

cJ

)

and P0

(

cJ

)

as the orthogonal projection onto

V(

^cJ

,

1

)

,

V(

^cJ

,

¹₂

)

and

V(

^cJ

,

0

)

, respectively. From [21], we know that

L(

^z

)

is positive definite on

V(

^cJ

,

1

)

and is a one-to-one mapping from

V(

^cJ

,

1

)

to

V(

^cJ

,

1

)

. This means that

L(

^z

)

an inverse

L

⁻¹

(

z

)

on

V(

^cJ

,

1

)

, i.e., for any u

∈ V(

^cJ

,

1

)

,

L

⁻¹

(

z

)

u is the unique v

∈ V(

^cJ

,

1

)

such that z

◦

^v

=

^u.

Lemma 3.3. For any x

,

y

∈ V

^{, let z}

: V × V → V

be the mapping defined as z

=

^z

(

x

,

y

) := 

x²

+

^y2

+ ( τ ₋

2

)(

x

◦

^y

) 

1/2

.

(25)

If

(

x

,

y

) = (

⁰

,

0

)

such that z

(

x

,

y

) / ∈

^int

( K)

, then the following results hold:

(a) The vectors x

,

y

,

x

+

^y

,

x

+

^τ−₂^{2y and y}

+

^τ−₂²x belong to the subspace

V(

^cJ

,

1

)

.

(b) For any h

∈ V

such that z²

(

x

,

y

) +

^h

∈ K

^{, let w}

=

^w

(

x

,

y

) := [

^z2

(

x

,

y

) +

^h

]

^1/2

−

^z

(

x

,

y

)

. Then, P1

(

cJ

)

w

=

¹₂

L

⁻¹

(

z

(

x

,

y

)) × [

^P1

(

cJ

)

h

] +

^o

(

^h

)

^.

Proof. From (19) and the definition of z, it is clear that z

(

x

,

y

) ∈ K

^{for all x}

,

y

∈ V

. Hence, using the similar arguments as Lemma 11 of [21] yields the desired result.

2

Now by Lemmas 3.2 and 3.3, we prove the differentiability of the merit function

ψ

τ .

Proposition 3.1. The function

ψ

τ defined by (6) is differentiable everywhere on

V × V

. Furthermore,

∇

x

ψ

τ

(

0

,

0

) = ∇

y

ψ

τ

(

0

,

0

) =

^0, and if

(

x

,

y

) = (

⁰

,

0

)

, then

∇

x

ψ

τ

(

x

,

y

) =

 L



x

+ τ −

² 2 y

 L

⁻¹



z

(

x

,

y

) 

− I

 φ

τ

(

x

,

y

),

∇

y

ψ

τ

(

x

,

y

) =

 L



y

+ τ −

² 2 x

 L

⁻¹



z

(

x

,

y

) 

− I



φ

τ

(

x

,

y

),

(26)

where z

(

x

,

y

)

is given by (25).

Proof. We prove the conclusion by the following three cases.

Case (1):

(

x

,

y

) = (

⁰

,

0

)

. For any u

,

v

∈ V

, suppose that u²

+

^v2

+ ( τ ₋

2

)(

u

◦

^v

)

has the spectrum decomposition u²

+

^v2

+ ( τ −

²

)(

u

◦

^v

) =

_r

j=¹

μ

jd_j, where

{

^d1

,

d₂

, . . . ,

d_r

}

is the corresponding Jordan frame. Then, for j

=

¹

,

2

, . . . ,

r, we have